Tate cohomology and local base change of generic representations of ${\rm GL}_3$ -- non-banal case

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite Galois extension of $F$ with degree of extension $l$, where $l$ and $p$ are distinct odd primes. Let $π_F$ be an integral, $l$-adic generic representation of ${\rm GL}_3(F)$, and let $π_E$ be the base change lifting of $π_F$ to ${\rm GL}_3(E)$. Let $J_l(π_F)$ (resp. $J_l(π_E)$) be the unique generic sub-quotient of the mod-$l$ reduction of $π_F$ (resp. $π_E$). In this article, using the local converse theorem over local Artinian $\overline{\mathbb{F}}_l$-algebras, we prove that the Frobenius twist of $J_l(π_F)$ is isomorphic to the Tate cohomology group $\widehat{H}^0({\rm Gal}(E/F),J_l(π_E))$. The result of this article removes the hypothesis that the prime $l$ does not divide the pro-order of ${\rm GL}_2(F)$.